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Solution of Radiative Transfer Problems with Finite Elements

  • Guido Kanschat

Summary

Mathematical modeling for monochromous radiative transfer problems is reviewed. Suitable boundary conditions for well-posed problems are introduced. Finite element discretizations for the integral operator in the angular variable and the transport operator in space are discussed. Adaptive algorithms and error estimates are explained. The structure of the resulting discrete linear system is analyzed and solution methods are suggested.

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References

  1. 1.
    W. Bangerth and R. Rannacher. Finite element approximation of the acoustic wave equation: Error control and mesh adaptation. East-West J. Numer. Math., 7(4):263–282, 1999. zbMATHMathSciNetGoogle Scholar
  2. 2.
    W. Bangerth and R. Rannacher. Adaptive Finite Element Methods for Solving Differential Equations. Birkhäuser, Basel, 2003. Google Scholar
  3. 3.
    R. Becker. An Adaptive Finite Element Method for the Incompressible Navier-Stokes Equations on Time-Dependent Domains. Dissertation, Universität Heidelberg, 1995. Google Scholar
  4. 4.
    R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, 2001. zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Brooks and T.J.R. Hughes. Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 32:199–259, 1982. zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, volume 6. Springer, 2000. Google Scholar
  7. 7.
    W. Dörfler. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal., 33:1106–1124, 1996. zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal., 28:43–77, 1991. zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J.P.d.S.R. Gago, D.W. Kelly, O.C. Zienkiewicz, and I. Babuška. A posteriori error analysis and adaptive processes in the finite element method: Part II—Adaptive mesh refinement. Internat. J. Numer. Methods Engrg., 19:1621–1656, 1983. zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Johnson. Finite Element Methods for Partial Differential Equations. Studentlitteratur, Lund, 1993. Google Scholar
  11. 11.
    C. Johnson, U. Nävert, and J. Pitkäranta. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45:285–312, 1984. zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    G. Kanschat. Parallel adaptive algorithms for radiative transfer problems. In P. Fritzson and L. Finmo, editors, Parallel Programming and Applications, volume 45 of Transputer and OCCAM Engineering Series, pages 238-243, Amsterdam, 1995. IOS Press. Google Scholar
  13. 13.
    G. Kanschat. Parallel and Adaptive Galerkin Methods for Radiative Transfer Problems. Dissertation, Universität Heidelberg, 1996. Preprint SFB 359, 1996-29. Google Scholar
  14. 14.
    G. Kanschat. Efficient and reliable solution of multi-dimensional radiative transfer problems. In F. Karsch, B. Monien, and H. Satz, editors, Multiscale Phenomena and Their Simulation, pages 245-249, Singapore, 1997. World Scientific. Google Scholar
  15. 15.
    G. Kanschat. New algorithms for radiative transfer in accretion disks and surroundings. In D.T. Wickramasinghe, G.V. Bicknell, and L. Ferrario, editors, Accretion Phenomena and Related Outflows; IAU Colloquium 163, pages 736-737, San Francisco, California, 1997. Astronomical Society of the Pacific. Google Scholar
  16. 16.
    G. Kanschat. A robust finite element discretization for radiative transfer problems with scattering. East-West J. Numer. Math., 6(4):265–272, 1998. zbMATHMathSciNetGoogle Scholar
  17. 17.
    G. Kanschat. Parallel computation of multi-dimensional neutron and photon transport in inhomogeneous media. In H.-J. Bungartz, F. Durst, and C. Zenger, editors, High Performance Scientific and Engineering Computing, volume 8, pages 431–440. Springer, 1999. Google Scholar
  18. 18.
    G. Kanschat. Solution of multi-dimensional radiative transfer problems on parallel computers. In P. Bjørstad and M. Luskin, editors, (eds.), Parallel Solution of Partial Differential Equations, volume 120 of IMA Volumes in Mathematics and its Applications, pages 85-96. New York, 2000. Springer. Google Scholar
  19. 19.
    G. Kanschat and E. Meinköhn. Multi-model preconditioning for radiative transfer problems. Preprint 2004-33, SFB 359, Heidelberg, 2004. submitted. Google Scholar
  20. 20.
    G. Kanschat and R. Rannacher. Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems. J. Numer. Math., 10(4):249–274, 2002. zbMATHMathSciNetGoogle Scholar
  21. 21.
    G. Kanschat and F.-T. Suttmeier. Datenstrukturen für die Methode der finiten Elemente. unpublished, Bonn-Venusberg, 1992. Google Scholar
  22. 22.
    G. Kanschat and F.-T. Suttmeier A posteriori error estimates for nonconforming finite element schemes. Calcolo, 36(3):129–141, 1999. zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    E.W. Larsen. Transport acceleration methods as two-level multigrid algorithms. In Modern Mathematical Methods in Transport Theory, pages 34–47, Basel, 1989. Birkhäuser. Google Scholar
  24. 24.
    P. LeSaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical aspects of finite elements in partial differential equations, pages 89–123, New York, 1974. Academic Press. Google Scholar
  25. 25.
    E. Meinköhn. A general-purpose finite element method for 3d radiative transfer problems in moving media. In this volume, 2005. Google Scholar
  26. 26.
    K.J. Ressel. Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes. PhD thesis, University of Colorado, Denver, 1994. Google Scholar
  27. 27.
    S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat. Radiative transfer with finite elements I. basic method and tests. A&A, 380:776–788, 2001. CrossRefGoogle Scholar
  28. 28.
    Y. Saad. Iterative Methods for Sparse Linear Systems. Oxford University Press, 2nd edition, 2000. Google Scholar
  29. 29.
    F.-T. Suttmeier. Adaptive Finite Element Approximation of Problems in Elasto-Plasticity Theory. Dissertation, Universität Heidelberg, 1996. Google Scholar
  30. 30.
    S. Turek. An efficient solution technique for the radiative transfer equation. Imp. Comput. Sci. Engrg., 5:201–214, 1993. zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    H. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 13(2):631–644, 1992. zbMATHCrossRefGoogle Scholar
  32. 32.
    R. Verfürth. A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math., 50:67–83, 1994. zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    R. Wehrse and G. Kanschat. Radiative fluxes and forces in non-spherical winds. In B. Wolf, O. Stahl, and A.W. Fullerton, editors, Variable and Non-spherical Stellar Winds in Luminous Hot Stars, pages 144–150. Springer, 1999. Google Scholar
  34. 34.
    R. Wehrse, E. Meinköhn, and G. Kanschat. A review of Heidelberg radiative transfer equation solutions. In P. Stee, editor, Radiative Transfer and Hydrodynamics in Astrophysics, pages 13–30. EDP Sciences, 2002. Google Scholar
  35. 35.
    K. Yosida. Functional Analysis. Springer, 1980. Google Scholar
  36. 36.
    H. Yserentant On the multi-level splitting of finite element spaces. Numer. Math., 49:379–412, 1986. zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    G. Zhou How accurate is the streamline diffusion method? Math. Comput., 66(217):31–44, 1997. zbMATHCrossRefGoogle Scholar
  38. 38.
    G. Zhou and R. Rannacher Pointwise superconvergence of the streamline diffusion finite element method. Numer. Meth. PDE, 12:123–145, 1996. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Guido Kanschat
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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